Rational embeddings of hyperbolic groups

TL;DR

This paper demonstrates that all Gromov hyperbolic groups can be embedded into a specific rational group using a novel boundary address system, linking geometric group theory with automata theory.

Contribution

It introduces a method to embed hyperbolic groups into the asynchronous rational group via boundary addresses and transducer actions, a new connection between geometric and automata groups.

Findings

All Gromov hyperbolic groups embed into the asynchronous rational group.

Addresses derived from a self-similar tree encode boundary points.

Group actions are realized through transducers on these addresses.

Abstract

We prove that all Gromov hyperbolic groups embed into the asynchronous rational group defined by Grigorchuk, Nekrashevych and Sushchanski\u{i}. The proof involves assigning a system of binary addresses to points in the Gromov boundary of $G$, and proving that elements of $G$ act on these addresses by transducers. These addresses derive from a certain self-similar tree of subsets of $G$, whose boundary is naturally homeomorphic to the horofunction boundary of $G$.